Skip to main content
SpringerLink
Log in

A Phase-Field Model for Articular Cartilage Regeneration in Degradable Scaffolds

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

Degradable scaffolds represent a promising solution for tissue engineering of damaged or degenerated articular cartilage which due to its avascular nature, is characterized by a low self-repair capacity. To estimate the articular cartilage regeneration process employing degradable scaffolds, we propose a mathematical model as the extension of Olson and Haider’s work (Int. J. Pure Appl. Math. 53:333–353, 2009). The simulated tissue engineering procedure consists in (i) the explant of a cylindrical sample, (ii) the removal of the inner core region, and (iii) the filling of the inner region with hydrogels, degradable scaffolds enriched with nutrients, such as oxygen and glucose. The phase-field model simulates the cartilage regeneration process at the scaffold-cartilage interface. It embeds reaction-diffusion equations, which are used to model the nutrient and regenerated extracellular matrix. The equations are solved using an unconditionally stable hybrid numerical scheme. Cartilage repair processes with full-thickness defects, which are controlled by properties of hydrogel materials and cartilage explant culture based on biological interest are observed. The implemented mathematical model shows the capability to simulate cartilage repairing processes, which can be virtually controlled evaluating hydrogel and cartilage material properties including nutrient supply and defected magnitude. In particular, the adopted methodology is able to explain the regeneration time of cartilage within hydrogel environments. With the numerical scheme, the numerical simulations are demonstrated for the potential improvement of hydrogel structures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  • Betre, H., Setton, L., Meyer, D., & Chilkoti, A. (2002). Characterization of a genetically engineered elastin-like polypeptide for cartilaginous tissue repair. Biomacromolecules, 3, 910–916.

    Article  Google Scholar 

  • Briggs, W. L. (1987). A multigrid tutorial. Philadelphia: SIAM.

    MATH  Google Scholar 

  • Burkitt, H. G., Young, B., & Heath, J. W. (1993). Functional histology (3rd ed.). Edinburgh: Churchill.

    Google Scholar 

  • Darling, E. M., & Athanasiou, K. A. (2003). Articular cartilage bioreactor and bioprocesses. Tissue Eng., 9, 9–26.

    Article  Google Scholar 

  • Davis, K. A., Burdick, J. S., & Anseth, K. S. (2003). Photoinitiated crosslinked degradable copolymer networks for tissue engineering applications. Biomaterials, 24, 2485–2495.

    Article  Google Scholar 

  • Elisseeff, J., Anseth, K., Sims, D., McIntosh, W., Randolph, M., Yaremchuk, M., & Langer, R. (1999). Transdermal photopolymerization of poly(ethylene oxide)-based injectable hydrogels for tissue-engineered cartilage. Plast. Reconstr. Surg., 104, 1014–1022.

    Article  Google Scholar 

  • Freed, L. E., Marquis, J. C., Langer, R., & Vunjak-Novakovic, G. (1994). Kinetics of chondrocyte growth in cell-polymer implants. Biotechnol. Bioeng., 43, 597–604.

    Article  Google Scholar 

  • Galban, C. J., & Locke, B. R. (1997). Analysis of cell growth in a polymer scaffold using a moving boundary approach. Biotechnol. Bioeng., 56, 422–432.

    Article  Google Scholar 

  • Gao, L.-T., Feng, X.-Q., & Gao, H. (2009). A phase field method for simulating morphological evolution of vesicles in electric fields. J. Comput. Phys., 228, 4162–4181.

    Article  MATH  Google Scholar 

  • Glimm, T., Zhang, J., Shen, Y.-Q., & Newman, S. A. (2012). Reaction-diffusion systems and external morphogen gradients: the two-dimensional case, with an application to skeletal pattern formation. Bull. Math. Biol., 74, 666–687.

    Article  MathSciNet  MATH  Google Scholar 

  • Grote, M. J., Palumberi, V., Wagner, B., Barbero, A., & Martin, I. (2011). Dynamic formation of oriented patches in chondrocyte cell cultures. J. Math. Biol., 63, 757–777.

    Article  MATH  Google Scholar 

  • Jackson, D. W., Lalor, P. A., Aberman, H. M., & Simon, T. M. (2001). Spontaneous repair of full-thickness defects of articular cartilage in a goat model. a preliminary study. J. Bone Jt. Surg. Am., 83, 53–64.

    Article  Google Scholar 

  • Kooi, B. W., Auger, P., & Poggiale, J. C. (1998). Aggregation methods in food chains. Math. Comput. Model., 27, 109–120.

    Article  MathSciNet  MATH  Google Scholar 

  • Kuo, Y., & Tsai, Y.-T. (2010). Inverted colloidal crystal scaffolds for uniform cartilage regeneration. Biomacromolecules, 11, 731–739.

    Article  Google Scholar 

  • Leddy, H. A., & Guilak, F. (2003). Site-specific molecular diffusion in articular cartilage measured using fluorescence recovery after photobleaching. Ann. Biomed. Eng., 31, 753–760.

    Article  Google Scholar 

  • Li, Y., Ma, T., Kniss, D. A., Lasky, L. C., & Yang, S. T. (2001). Effects of filtration seeding on cell density, spatial distribution, and proliferation in nonwoven fibrous matrices. Biotechnol. Prog., 17(5), 935–944.

    Article  Google Scholar 

  • Li, Y., Lee, H. G., Jeong, D., & Kim, J. S. (2010). An unconditionally stable hybrid numerical method for solving the Allen–Cahn equation. Comput. Math. Appl., 60, 1591–1606.

    Article  MathSciNet  MATH  Google Scholar 

  • Murray, J. D. (2002). Mathematical biology. Berlin: Springer.

    MATH  Google Scholar 

  • Min, C. (2010). On reinitializing level set functions. J. Comput. Phys., 229, 2764–2772.

    Article  MathSciNet  MATH  Google Scholar 

  • Nettles, D. L., Parker Vail, T., Morgan, M., Grinstaff, M., & Setton, L. (2004). Photocrosslinkable hyaluronan as a scaffold for articular cartilage repair. Ann. Biomed. Eng., 32, 391–397.

    Article  Google Scholar 

  • Nguyen, K. T., & West, J. L. (2002). Photopolymerizable hydrogels for tissue engineering applications. Biomaterials, 23, 4307–4314.

    Article  Google Scholar 

  • Noguchi, T., Yamamuro, T., Oka, M., Kumar, P., Kotoura, Y., Hyonyt, S. H., & Ikadat, Y. (1991). Poly (vinyl alcohol) hydrogel as an artificial articular cartilage: evaluation of biocompatibility. J. Appl. Biomater., 2, 101–107.

    Article  Google Scholar 

  • Olson, S. D., & Haider, M. A. (2009). A level set reaction-diffusion model for tissue regeneration in a cartilage-hydrogel aggregate. Int. J. Pure Appl. Math., 53, 333–353.

    MathSciNet  MATH  Google Scholar 

  • Ossendorf, C., Kaps, C., Kreuz, P., Burmester, G., Sittinger, M., & Erggelet, C. (2007). Treatment of posttraumatic and focal osteoarthritic cartilage defects of the knee with autologous polymer-based three-dimensional chondrocyte grafts: 2-year clinical results. Arthritis Res. Ther., 9, R41.

    Article  Google Scholar 

  • Rydholm, A. E., Bowman, C. N., & Anseth, K. S. (2005). Degradable thiol-acrylate photopolymers: polymerization and degradation behavior of an in situ forming biomaterial. Biomaterials, 26, 4495–4906.

    Article  Google Scholar 

  • Sanz-Herrera, J. A., Garcia-Aznar, J. M., & Doblare, M. (2009). On scaffold designing for bone regeneration: a computational multiscale approach. Acta Biomater., 5, 19–29.

    Google Scholar 

  • Sengers, B. G., Taylor, M., Please, C. P., & Oreffo, R. O. C. (2007). Computational modelling of cell spreading and tissue regeneration in porous scaffolds. Biomaterials, 28(10), 1926–1940.

    Article  Google Scholar 

  • Shin, J., Jeong, D., & Kim, J. S. (2011). A conservative numerical method for the Cahn–Hilliard equation in complex domains. J. Comput. Phys., 230, 7441–7455.

    Article  MathSciNet  MATH  Google Scholar 

  • Shakeel, M., Matthews, P. C., Graham, R. S., & Waters, S. L. (2013). A continuum model of cell proliferation and nutrient transport in a perfusion bioreactor. Math. Med. Biol., 30, 21–44.

    Article  MathSciNet  MATH  Google Scholar 

  • Shirakawa, K., & Kimura, M. (2005). Stability analysis for Allen–Cahn type equation associated with the total variation energy. Nonlinear Anal., 60, 257–282.

    Article  MathSciNet  MATH  Google Scholar 

  • Stile, R. A., Burghardt, W. R., & Healy, K. E. (1999). Synthesis and characterization of injectable poly(N-isopropylacrylamide)-based hydrogels that support tissue formation in vitro. Macromolecules, 32, 7370–7379.

    Article  Google Scholar 

  • Taylor, J. E., & Cahn, J. W. (1998). Diffuse interfaces with sharp corners and facets: phase field models with strongly anisotropic surfaces. Physica D, 112, 381–411.

    Article  MathSciNet  MATH  Google Scholar 

  • Temenoff, J. S., & Mikos, A. G. (2000). Review: tissue engineering for regeneration of articular cartilage. Biomaterials, 21, 431–440.

    Article  Google Scholar 

  • Trewenack, A. J., Please, C. P., & Landman, K. A. (2009). A continuum model for the development of tissue-engineered cartilage around a chondrocyte. Math. Med. Biol., 26(3), 241–262.

    Article  MATH  Google Scholar 

  • Trottenberg, U., Oosterlee, C., & Schüller, A. (2001). Multigrid. San Diego: Academic Press.

    MATH  Google Scholar 

  • Wilson, C. G., Bonassar, L. J., & Kohles, S. S. (2002). Modeling the dynamic composition of engineered cartilage. Arch. Biochem. Biophys., 408, 246–254.

    Article  Google Scholar 

  • Wilson, D. J., King, J. R., & Byrne, H. M. (2007). Modelling scaffold occupation by a growing, nutrient-rich tissue. Math. Models Methods Appl. Sci., 17S, 1721–1750.

    Article  MathSciNet  MATH  Google Scholar 

  • Yang, X., Feng, J. J., Liu, C., & Shen, J. (2006). Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method. J. Comput. Phys., 218, 417–428.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author (A. Yun) was supported by National Junior research fellowship from the National Research Foundation of Korea grant funded by the Korea government (No. 2011-00012258). The corresponding author (J.S. Kim) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0027580). The authors also wish to thank the anonymous referee for the constructive and helpful comments on the revision of this article.

Author information

Authors and Affiliations

  1. Department of Mathematics, Korea University, Seoul, 136-713, Republic of Korea

    Ana Yun & Junseok Kim

  2. Department of Orthopedic Surgery, Korea University College of Medicine, Seoul, 136-705, Republic of Korea

    Soon-Hyuck Lee

Authors
  1. Ana Yun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Soon-Hyuck Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Junseok Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junseok Kim.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yun, A., Lee, SH. & Kim, J. A Phase-Field Model for Articular Cartilage Regeneration in Degradable Scaffolds. Bull Math Biol 75, 2389–2409 (2013). https://doi.org/10.1007/s11538-013-9897-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-013-9897-3

Keywords

Search

Navigation